Properties of Stock Options
Properties of Stock Options
Chapter 9
1 Options, Futures, and Other
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull
2008 2
Notation
Notation
c
:
European call
option price
p
:
European put
option price
S
0
:
Stock price
today
K
:
Strike price
T
:
Life of option
:
Volatility of
stock price
C
:
American Call
option price
P
:
American Put
option price
S
T:Stock price at option
maturity
D
:
Present value of
dividends during option’s
life
r
:
Risk-free rate for
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull
2008 3
Effect of Variables on Option
Effect of Variables on Option
Pricing
Pricing
(Table 9.1, page 202)
(Table 9.1, page 202)
c
p
C
P
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull
2008 4
American vs European Options
American vs European Options
An American option is worth at
least as much as the
corresponding European option
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull
2008 5
Calls: An Arbitrage
Calls: An Arbitrage
Opportunity?
Opportunity?
Suppose that
c
= 3
S
0
= 20
T
= 1
r
= 10%
K
= 18
D
= 0
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull
2008 6
Lower Bound for European
Lower Bound for European
Call Option Prices; No
Call Option Prices; No
Dividends (
Dividends (
Equation 9.1, page 207)
Equation 9.1, page 207)
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull
2008 7
Puts: An Arbitrage
Puts: An Arbitrage
Opportunity?
Opportunity?
Suppose that
Is there an arbitrage
opportunity?
p
= 1
S
0
= 37
T
= 0.5
r
=5%
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull
2008 8
Lower Bound for European Put
Lower Bound for European Put
Prices; No Dividends
Prices; No Dividends
(Equation 9.2, page 208)
(Equation 9.2, page 208)
p
max
(
Ke
-rT
–S
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull
2008 9
Put-Call Parity; No Dividends
Put-Call Parity; No Dividends
(Equation 9.3, page 208)
(Equation 9.3, page 208)
Consider the following 2 portfolios:
◦
Portfolio A: European call on a stock + PV of the
strike price in cash
◦
Portfolio C: European put on the stock + the
stock
Both are worth max(
S
T
,
K
) at the maturity of the
options
They must therefore be worth the same today. This
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull
2008 10
Arbitrage Opportunities
Suppose that
c
= 3
S
0
= 31
T
= 0.25
r
= 10%
K
= 30
D
= 0
What are the arbitrage
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull
2008 11
Early Exercise
Early Exercise
Usually there is some chance that an
American option will be exercised early
An exception is an American call on a
non-dividend paying stock
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull
2008 12
For an American call option:
S
0
= 100;
T
= 0.25;
K
= 60;
D
= 0
Should you exercise immediately?
What should you do if
◦
you want to hold the stock for the next 3 months?
◦
you do not feel that the stock is worth holding for
the next 3 months?
An Extreme Situation
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull
2008 13
Reasons For Not Exercising
Reasons For Not Exercising
a Call Early (No Dividends)
a Call Early (No Dividends)
No income is sacrificed
Payment of the strike price is delayed
Holding the call provides insurance
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull
2008 14
Should Puts Be Exercised
Should Puts Be Exercised
Early ?
Early ?
Are there any advantages to exercising
an American put when
S
0
= 60;
T
= 0.25;
r
=10%
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull
2008 15
The Impact of Dividends on
The Impact of Dividends on
Lower Bounds to Option Prices
Lower Bounds to Option Prices
(Equations 9.5 and 9.6, pages 21
(Equations 9.5 and 9.6, pages 21
4
4
-2
-2
15
15
)
)
rT
Ke
D
S
c
0
0
S
Ke
D
Options, Futures, and Other Derivatives 7th Edition, Copyright © John C. Hull
2008 16
Extensions of Put-Call Parity
Extensions of Put-Call Parity
American options;
D
=
0
S
0
-
K
<
C
-
P
<
S
0
-
Ke
-rT
(Equation 9.4, p. 211)
European options;
D
>
0
c
+
D
+
Ke
-rT
=
p
+
S
0
(Equation 9.7, p. 215)
American options;
D
>
0
S
0
-
D
-
K
<
C
-
P
<
S
0
-
Ke
-rT